Existing methods for estimating nonlinear dynamic models are either too computationally complex to be of practical use, or rely on local approximations which fail to adequately capture the nonlinear features of interest. I develop a new method, the discretization filter, for approximating the likelihood of nonlinear, non-Gaussian state space models. I apply results from the statistics literature on uniformly ergodic Markov chains to establish that the implied maximum likelihood estimator is strongly consistent, asymptotically normal, and asymptotically efficient. Through simulations I show that the discretization filter is orders of magnitude faster than alternative nonlinear techniques for the same level of approximation error and I provide practical guidelines for applied researchers. I apply my approach to estimate two models at the intersection of macroeconomics and finance: the Wu and Xia (2016) shadow rate term structure model, and the Gabaix (2012) asset pricing model of variable rare disasters. I provide the first estimates of the Gabaix model and use them to construct measures of disaster risk for the U.S. economy. My estimates of the Wu and Xia shadow rate indicate that unconventional monetary policy was more effective than previously thought.